Hello,
Chris Freiling’s Axioms of Symmetry have, I believe, been discussed on
FOM at least twice. In ‘Axioms of Symmetry: Throwing Darts at the Real
Number Line’, Freiling considers two darts thrown at [0, 1]. He
writes, ‘the real number line does not really know which dart was
thrown first or second’, which leads to one of his axioms of symmetry.
In a recently published paper, I suggest that a well-ordering of [0,
1] does know the order of the darts under certain assumptions. Fix a
well-ordering of [0, 1]. Let r1 be the real hit by the first dart.
Then assuming ZFC and CH, there are only countably many reals less
than r1 in the well-ordering. Thus with probability 1 the second dart
hits a real greater than r1 in the well-ordering. (Put again slightly
differently: working in ZFC, Freiling demonstrates that assuming that
the reals can’t tell the order of the darts proves not CH; I argue
that assuming CH means that a well-ordering of [0, 1] can tell the
order of the darts.) I go on to create a puzzle using special
relativity. In case it is of interest, the paper is here:
http://www.springerlink.com/content/e2746w1kn6913580/
An earlier, countable version of a similar puzzle is available here:
http://www.springerlink.com/content/v1n12t200jv2553u/
Best,
Jeremy Gwiazda
jgwiazda at gc.cuny.edu